REVIEW 3 major objections 7 minor 118 references
Classical simulation and model concentration in passive linear optics
T0 review · 3 major / 7 minor · reviewed 2026-07-31 · grok-4.5
Pith's one-line read In passive linear optics, barren plateaus and classical simulability are linked by how much the input state and observable share weight on the same irreducible pieces of the unitary group.
desk verdict Solid regime map for bosonic barren plateaus built on their prior second-moment calculus; the advertised partial separation rests on thin finite-n fits the paper itself half-walks back. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The irrep-purity variance formula (Theorem 1): Var = Σ_k ||P_k(ρ)||₂² ||P_k(O)||₂² / d_k, with purities obtained from an iterative lowering-map expansion of partial traces. It turns generalized entanglement (state side) and generalized locality (observable side) into concrete asymptotic weights that both diagnose concentration and build average-case classical surrogates by truncating large-k tails.
What would settle it
Compute or rigorously bound the high-irrep tail of the variance for the nonlinear number-phase observable on minimally bunched Fock inputs at substantially larger n (or prove a matching worst-case classical algorithm); if the tail becomes exponentially small, or if a poly-time worst-case estimator appears, the claimed partial separation disappears.
Extended reading notes
Core claim
Concentration of expectation values under random passive linear-optical circuits is governed by misalignment of the projections of the input state and the observable into irreducible representations of U(m): the Haar variance equals the sum over k of ||P_k(ρ)||₂² ||P_k(O)||₂² / d_k. Exponential concentration fails only when state and observable share non-negligible weight on irreps whose normalized projections compensate dimension; conversely, many non-concentrating observables remain classically simulable, and the paper's best candidate separations still leave a high-irrep tail small enough that truncation is a classical surrogate with polynomially small error.
Load-bearing premise
That finite-size numerical fits up to a few dozen photons, plus the existing catalogue of classical algorithms, are enough to decide asymptotic concentration and to claim that residual high-irrep signal is beyond known efficient simulation while still admitting a polynomially accurate classical surrogate.
Editorial extensions
If this is right
- Constant-degree photon-number products and single-support monomials avoid exponential concentration only by living in polynomially large irreps and are exactly classically simulable.
- Fock projectors and many growing-degree number products concentrate exponentially because state and observable irrep weights misalign or cannot beat dimension.
- Nonlinear number-phase observables can keep a non-exponentially vanishing high-irrep contribution, but low-irrep truncation still approximates the landscape to polynomial relative error.
- The same purity pipeline is a constructive search method for bosonic observables that might simultaneously avoid barren plateaus and known efficient classical simulation.
- Dimensional bounds alone forbid non-negligible signal from irreps with k/n above a fixed threshold set by the mode-to-photon ratio.
Reading between the lines
- A true trainability–simulability separation in linear optics likely requires observables whose high-irrep weight is a constant fraction of the full variance, not a large inverse polynomial.
- Extending the same irrep-purity diagnosis to adaptive or nonlinear optics would test whether the partial-separation pattern is special to the passive unitary group.
- The necessity that state purity on mid-scale irreps be at least exponentially small yet compensable by observable weight suggests designing resource states deliberately peaked near the dimensional crossover rather than at extreme bunching.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies concentration of expectation values of particle-number-preserving observables on Fock-state inputs under Haar-random passive linear-optical interferometers. Using the irrep decomposition of the n-photon operator space W_n under the adjoint U(m) action, it recalls (from companion work [1]) a variance formula Var = Σ_k ||P_k(ρ)||²₂ ||P_k(O)||²₂ / d_k (Theorem 1) and an iterative procedure to evaluate the irrep purities in closed form. It analyzes Fock-state purity profiles as a function of bunching (Lemma 2, Figs. 2–3), derives support rules for monomial observables (Propositions 1–3), reviews five classical simulation techniques, and classifies concrete observable families into four regimes (VB1–VB4) combining concentration behavior with simulability. The headline claim is deliberately modest: number-phase observables with minimally bunched inputs appear to avoid exponential concentration with a non-negligible high-irrep component not covered by known worst-case algorithms, but the residual signal is small enough that low-irrep truncation is a classical surrogate with polynomially small error — a 'partial separation' only.
Significance. If the claims hold, the paper provides the first systematic representation-theoretic treatment of concentration and its relation to classical simulability in passive linear optics, extending the qubit barren-plateau/simulability tradeoff to a bosonic setting where the 'curse of dimensionality' mechanism can genuinely differ. Concrete strengths: closed-form variance and irrep-purity formulas with explicit iterative evaluation procedures (adapted from [1]); provable construction rules linking observable monomial structure to irrep support; a clear formulation of the signal-relative truncation error ε_rel(K) (Eq. (33)) that makes the notion of 'classical surrogate' precise; an honest discussion of the max-bunched input being classically simulable via a multinomial/dynamic-programming argument (App. G3); and falsifiable numerical predictions with code promised. The four-regime taxonomy (VB1–VB4) is a useful organizational contribution even where its boundary cases remain open.
major comments (3)
- [Table IV; §VB4; Table I] Table IV and the paragraph following it (App. G4), feeding into the orange-highlighted cell of Table I and regime VB4 in §VB4: in the decisive case — minimally bunched input, quadratic number-phase observable — the paper's own R² criterion prefers the exponential fit over the power-law fit for the truncated tail at k0=⌊√n⌋ (R²_E−R²_P ≈ 0.018 for p=2, 0.024 for p=⌊log₂n⌋, 0.024 for p=⌊√n⌋), and at k0=⌊log₂n⌋ the power-law advantage is marginal (≤0.014). The statement 'the truncated signals do not vanish exponentially for the large truncation orders' is therefore not supported by the reported fits; by the paper's own bolding convention the quadratic/min-bunched truncated cells favor exponential decay. If the high-irrep tail is exponentially small, this example belongs to VB2 (average-case g-simulation surrogate), not VB4, and the partial-separation narrative of the abstract and Fig. 1 lose
- [App. E5, F3, G4; Eq. (34)] Range of the numerics and cutoff structure: n runs from 2 to 35 (m=2n), so the 'growing' cutoffs k0=⌊log₂n⌋ and k0=⌊√n⌋ never exceed 5. Fitting a two-parameter power law and a two-parameter exponential to ≤34 points in log space and comparing R² values is weak evidence for an asymptotic classification; several fitted exponents (e.g. n^−29.66 in Table III) suggest the fits mix regimes. More importantly, the criterion in Eq. (34) requires the tail beyond every efficient truncation to remain non-negligible, but the tables only test moving cutoffs; a fixed efficient cutoff (say K=10) is never examined, so nothing bounds the tail between K=10 and ⌊√n⌋. A fixed-K column in Tables II–V, and ideally analytic estimates of Δ_K using the closed-form purities of Lemma 2 and App. G2, would substantially strengthen the load-bearing empirical claims.
- [§IIIC, Proposition 3; App. B1, Proposition 9] Proposition 3 (and its restatement as Proposition 9 in App. B): the statement is internally inconsistent — it says '|𝒑∩𝒒|=k, i.e., 𝒑 and 𝒒 have exactly k−1 common indices', and asserts support on 'the K consecutive irreps' with K undefined (k elsewhere). This proposition is used in the main text to justify the irrep support of Fock-state projectors (§VA, Eq. (42)) and the design rules of §IIIC, so the off-by-one/indexing confusion should be corrected and the boundary cases (k=1, k=d) stated explicitly and checked against Propositions 1 and 2.
minor comments (7)
- [§VA, Eq. (41); App. G1] The notation O=(−1)^{g_p({n̂_i})} (Eq. (41) and Eq. (G4)) is ambiguous for the quadratic case: with α_i=1/r_i, the exponent α_i n_i² is generally non-integer and (−1)^x is multi-valued. The intended object, e^{iπ n̂²/r} as in Eq. (G8), is well defined; the main text should define the observable directly in exponential form and note that the Hermitian observable is (O+O†)/2.
- [§IV, Eq. (33)] Eq. (33): ε_sim(K) is stated as Θ(Δ_K), but Eq. (32) gives exact equality of the mean-squared error with Δ_K; the Θ notation obscures that ε_rel has a closed form. Please clarify.
- [Code Availability] The Code Availability statement says the code is 'available on the following repository' but no URL or identifier is given. Since the empirical classifications in Tables II–V rest entirely on these numerics, a working repository link (ideally with the fitting scripts) is important for verification.
- [Various] Several typos and language issues: 'Clebsh–Gordan' (§IIIA), 'Obervable-dependent' (§IIIC heading), 'This result allow to retrieve' (§IV), 'non of the aforementioned' (§VB), 'the expression the the coefficients' (§IIIA), 'extensivelystudied' (§VA), and missing spaces throughout (e.g., 'remainslargelyunexplored' in the abstract).
- [Fig. 2; §II] Fig. 2: panels (a)–(c) would benefit from stating in the caption that the curves are fits/interpolations of the closed-form purities from Lemma 2, and from specifying the m/n ratio used (2n vs ⌈2.1n⌉); the two conventions are used in different places without comment.
- [App. G3; Table I] The maximally-bunched classical algorithm of App. G3 (multinomial sampling / O(pn²) dynamic program, generalizing to constantly many occupied modes) is a nice observation but is only mentioned in passing in §VB4; given that it eliminates the max-bunched row from any separation claim, it deserves a pointer from Table I.
- [§VB1; App. E5] §VB1, final paragraph: the statement that p=Θ(log n) variance fits are 'not conclusive since the R² scores were close' is the correct instinct and should be applied uniformly — see major comment 1. A short methods paragraph on fit uncertainty (e.g., confidence intervals on exponents, sensitivity to dropping small-n points) would help all tables.
Circularity Check
No significant circularity: variance identities follow from Haar/Schur math; self-cite [1] is shared infrastructure, not a fitted prediction renamed as result.
-
self citation load bearing
[Sec. IIIA, Theorem 1 and Theorem 2; also Abstract / Intro framing on [1]]
"The first- and second-moment formulas were formally derived in Ref. [1, Proposition 2]. ... In [1], we showed that these norms can be expressed as a linear combination of partial trace of the operator in question over particles. Theorem 2 (Informal, adapted from [1])."
The paper’s entire concentration analysis is computed inside the second-moment / iterative-projection apparatus of overlapping-author Ref. [1]. That is load-bearing shared infrastructure, not an external uniqueness theorem or a fitted constant renamed as prediction. Once the identities are granted, subsequent variance scalings and simulation comparisons are independent evaluations, so this raises the score only mildly.
full rationale
The load-bearing variance formula (Theorem 1 / Eq. 18) is the standard second-moment consequence of Haar invariance plus Schur orthogonality of U(m) irreps on W_n; it is not defined in terms of the concentration conclusions it is used to diagnose. Irrep-purity evaluations (Theorem 2 / iterative L/R maps) are closed-form algebraic identities adapted from overlapping-author work [1], but they are parameter-free procedures with stated assumptions, not fits of the target scaling. Regime classification (VB1–VB4, Table I) and the partial-separation narrative for number-phase observables are obtained by numerically evaluating those identities against an external catalogue of simulation algorithms (Gurvits, Ryser/Barvinok, Lim–Oh, g-sim). E[|f_U−f_C|²]=Δ_K is equality by construction of the irrep truncation, which the paper states as a method rather than as an independent empirical prediction. Weaknesses in finite-n R² fits (whether high-irrep tails are polynomial or exponential) are correctness/asymptotics risks, not circular reductions of claim to input. Score 1 only for mild foundational dependence on self-cited [1]; central concentration–simulability content does not collapse to a fit or a uniqueness import.
Assumptions & free parameters
free parameters (3)
- m = ⌈2.1 n⌉ (numerics often m = 2n) =
2.1 (or 2 in figures)
- α_i = 1/r_i (r_i = i-th prime) in nonlinear number-phase O =
1/prime_i
- Irrep cutoffs K ∈ {1, ⌊log₂ n⌋, ⌊√n⌋} and p ∈ {Θ(1), log n, √n}
assumptions (6)
- domain assumption Haar measure on U(m) is the right average-case ensemble for random passive interferometers and for diagnosing barren plateaus in photonic VQAs.
- standard math Second-moment formula Var = Σ_k ||P_k(ρ)||₂² ||P_k(O)||₂² / d_k from Schur and the irrep decomposition of W_n (Theorem 1 / Ref. [1]).
- standard math Bosonic CCR and particle-number-preserving algebra W spanned by equal-degree normally ordered monomials.
- domain assumption Catalogue of classical methods (exact/approx g-sim, Lim–Oh product estimators, Gurvits-style amplitudes, Ryser/Barvinok direct expansion, Fourier of phase shifters) exhausts “known efficient” simulation for the compared regimes.
- domain assumption Observables normalized to ||O||_∞ = O(1) so variance is not inflated by trivial rescaling against shot noise.
- ad hoc to paper Finite-n (n≤35, some plots to ~130) power-law vs exponential R² comparisons diagnose asymptotic concentration class.
invented entities (2)
-
Irrep purities ||P_k(X)||₂² as generalized entanglement (states) and generalized locality (observables)
-
Nonlinear number-phase / prime-weighted Kerr product observables as candidate “large-irrep signal” losses
independent evidence
Cite this review
Pith. "Pith review of Classical simulation and model concentration in passive linear optics." pith.science (2026). https://pith.science/paper/VRGPVEY4
@misc{pith2026260724728,
author = {Pith},
title = {Pith review of: Classical simulation and model concentration in passive linear optics},
year = {2026},
howpublished = {\url{https://pith.science/paper/VRGPVEY4}},
note = {Machine review of arXiv:2607.24728}
}
read the original abstract
Passive linear optics is a restricted model of quantum computation, with complexity-theoretic evidence of quantum advantage for sampling tasks and low losses that make it attractive for near-term algorithms. In qubit architectures, a body of work has revealed a close connection between barren plateaus and classical simulability. Whether an analogous tradeoff exists for bosonic systems remains largely unexplored. Building on a recently developed representation-theoretic framework for moments of random passive linear-optical circuits, we characterize the concentration of expectation values for relevant families of particle-number-preserving observables by evaluating their projections into irreducible representations of the unitary group and analyzing their asymptotic scaling. We show that concentration is governed by the misalignment of the projections into irreducible representations of the input state and the observable, giving a unified representation-theoretic interpretation of generalized entanglement and locality in the bosonic setting. We further relate these concentration properties to existing classical simulation techniques, identifying broad classes of trainable observables that admit efficient classical simulation. Conversely, we identify Fock-state inputs and observables that appear to evade exponential concentration while retaining a polynomially large signal component not accessible to known efficient classical simulation methods. The separation is only partial: most of the signal remains classically tractable, and the residual part, while not exponentially suppressed, is small enough that a truncation serves as a classical surrogate with polynomially small error. Our framework nonetheless provides a systematic route for searching for regimes that unambiguously combine the absence of exponential concentration and lies beyond known efficient classical simulation methods.
Figures
Figures from the paper (14 more)
Reference graph
Works this paper leans on
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If Supp(𝑂)⊆{0,...,𝐾},(45) then𝒱 >𝐾 =0, and the second moment is entirely de- termined by the small-irrep contribution
No concentration and exactly simulable The simplest way to avoid exponential concentra- tion is to restrict the observable exactly to polynomial- dimensional irreps. If Supp(𝑂)⊆{0,...,𝐾},(45) then𝒱 >𝐾 =0, and the second moment is entirely de- termined by the small-irrep contribution. Provided that 𝒱⩽𝐾 is not exponentially small, the model avoids expo- nen...
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No concentration due to signal from "small" irreps Anobservablemayhavesupportonlargeirrepswhile receiving a non-negligible contribution only from the small ones. In this case, 𝒱⩽𝐾 ∉𝒪(exp(−𝑛)),𝒱 >𝐾∈𝒪(exp(−𝑛)).(46) The model avoids exponential concentration because of itslow-irrepsignal,eventhoughtheobservableisnotex- actly supported on that sector. This re...
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No concentration and signal from "large" irreps The most interesting regime occurs when a non- negligible contribution survives on irreps beyond every efficient low-irrep truncation: 𝒱>𝐾 ∉𝒪(exp(−𝑛)).(48) While the contribution termΓ𝑘(𝜌,𝑂)cannot avoid ex- ponential concentration for𝑘>𝑛/2by simple dimen- sionalityargumentsdevelopedinSectionH,signalfrom irre...
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Proof.It suffices to show that there exists[𝑋](𝑑) 𝑑 ∈𝑊 𝑑 such that[𝑀 𝐼,𝐽](𝑛) 𝑑 =𝑅 𝑛...𝑅 𝑑+1([𝑋](𝑑) 𝑑 ), meaning that the monomial has zero support on the top𝑛−𝑑irreps
Irrep support construction proofs Proposition7.Amonomial[𝑀 𝒑,𝒒](𝑛) 𝑑 ofdegree𝑑⩽𝑛actingon𝑛particleshassupportonatmostthefirst𝑑+1irrepsof𝑊 𝑛, namely𝜆(𝑛) 0 ,...,𝜆 (𝑛) 𝑑 . Proof.It suffices to show that there exists[𝑋](𝑑) 𝑑 ∈𝑊 𝑑 such that[𝑀 𝐼,𝐽](𝑛) 𝑑 =𝑅 𝑛...𝑅 𝑑+1([𝑋](𝑑) 𝑑 ), meani...
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[102]
First, based on Eq
Proof of Simulation Technique 5 In order to prove Simulation Technique 5, we present different simulation techniques. First, based on Eq. (C1), we bound the number of summands required to approximate𝑓𝑈(|𝑆⟩⟨𝑆|,𝑀 𝒑,𝒒)via direct expansion. Proposition10(Cardinalityofthesummand).L...
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[103]
ConsideraFockstate𝑅withatotalnumberof𝑛photons,i.e𝑅∈Φ 𝑛 𝑚
Irrep purities of Fock states Lemma4.(IrreppuritiesforFockstates,adaptedfrom[1]). ConsideraFockstate𝑅withatotalnumberof𝑛photons,i.e𝑅∈Φ 𝑛 𝑚. For every0⩽𝑘⩽𝑛, the term𝑔 𝑘(|𝑅⟩⟨𝑅|)defined in Eq.(A3)can be expressed as 𝑔𝑘(|𝑅⟩⟨𝑅|)=((𝑛−𝑘)!) 2 Õ |𝒃|=𝑛−𝑘 0⩽𝒃⩽𝑅 𝑚Ö 𝑖=1 𝑅𝑖 𝑏𝑖 2 .(D1) The𝑘−...
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[104]
Irrep profile discussion Weproposeadditionalsimulationsandcommentsforthepuritydistributionofinitialstateinpassivelinearoptics. First, we observe that the distribution of Fock state|𝑅⟩ irrep purities is determined by the occupation integer string 𝑅=(𝑅 1,...,𝑅 𝑚)such that𝑅∈Φ 𝑚,𝑛...
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[105]
Definition and properties: We consider the case of product of photon number operators: 𝑂= Ö 𝑖∈𝐼 ˆ𝑛𝑖 = Õ 𝑅∈Φ𝑚,𝑛 Ö 𝑖∈𝐼 𝑅𝑖 ! |𝑅⟩⟨𝑅| (E1) The2-norm of this observable is given by: ||𝑂|| 2 2= Õ 𝑅∈Φ𝑚,𝑛 Ö 𝑖∈𝐼 𝑅𝑖 ! 2 (E2) And its infinite norm is given by: ||𝑂||∞=max 𝑅∈Φ𝑚,𝑛 Ö 𝑖∈𝐼 𝑅𝑖 ! (E3)
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[106]
Irrep Decomposition: The𝑘-application of the lowering map𝐿𝑘, introduced in Eq. (A1), on this operator is given by: 𝐿𝑘(𝑂)= Õ 𝑅∈Φ𝑚,𝑛 ( Ö 𝑖∈𝐼 𝑅𝑗) Õ |𝑏|=𝑘,𝑏⩽𝑅 𝑘! 𝑏1!...𝑏 𝑚! 𝑚Ö 𝑙=1 𝑅𝑙! (𝑅𝑙−𝑏 𝑙)! ! |𝑅−𝑏⟩⟨𝑅−𝑏| (E4) 29 𝑛=10 𝑛=20 𝑛=300 0.2 0.4 0.6 0.8 1 (arg max 𝑘⩽𝑛 ∥𝑃𝑘(|𝑅⟩⟨𝑅|)∥2 2)/𝑛 ...
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[107]
In addition, such observable can be written as a sum of monomial terms𝑀𝒑,𝒒 of degree𝑝
Classical Simulation Duetothefactthataproductof𝑝⩽𝑛paritynumberoperatorisanobservableofdegree𝑝,andthusonlyhassupport onthe𝑝+1firstirrep,apassivelinearopticsmodelbasedonsuchobservablecanalwaysbesimulateclassicallyusing Simulation Technique 1. In addition, such observable can be ...
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[108]
First Moment Studies: We consider the first moment of the observable expectation value, for different values of𝑝. 0 5 10152025303510−3 10−2 10−1 𝑛 E[𝑓𝑈] 𝑝=2 ˆ𝑛𝑝1Î𝑝𝑖=1ˆ𝑛𝑖 0 5 10152025303510−6 10−5 10−4 10−3 10−2 10−1 𝑛 𝑝=⌊log2𝑛⌋ ˆ𝑛𝑝1Î𝑝𝑖=1ˆ𝑛𝑖 0 5 10152025303510−6 10−5 10−4 10−3 ...
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[109]
(18), and recalled here: E 𝑈∼𝑈(𝑚) 𝑓𝑈(𝜌,𝑂) 2 = 𝑛Õ 𝑘=0 ∥𝑃(𝑛) 𝑘 (𝜌)∥2 2∥𝑃(𝑛) 𝑘 (𝑂)∥ 2 2 𝑑(𝑛) 𝑘
Second Moment Studies: To study the second moment, we propose to verify numerically the decay of the second moment given in Eq. (18), and recalled here: E 𝑈∼𝑈(𝑚) 𝑓𝑈(𝜌,𝑂) 2 = 𝑛Õ 𝑘=0 ∥𝑃(𝑛) 𝑘 (𝜌)∥2 2∥𝑃(𝑛) 𝑘 (𝑂)∥ 2 2 𝑑(𝑛) 𝑘 . In order to make sure that a polynomial decay is not on...
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[110]
This difference mainly comes from the fact that Fock states have mainlysupportonthelastirreps(seeSectionD)whileaproductof𝑝⩽𝑛photonnumberoperatorsonlyhavesupport onthefirst𝑝+1irreps
Cause of Barren Plateaus We notice that, in the particular case of a product of photon number operators, the presence of Barren Plateaus when𝑝=Ω(log 2𝑛)isduetothefactthatthepuritydistributionoftheinputstate{𝑃 (𝑛) 𝑘 (𝜌)}𝑛 𝑘=0 isnotalignedwiththe purity distribution of the obser...
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[111]
Definition and properties: Based on Proposition 2, we now that a monomial[𝑀𝒑,𝒒](𝑛) 𝑑 of degree𝑑⩽𝑛acting on𝑛particles such that𝑝 𝑖 ≠ 𝑝𝑗 ,∀𝑖,𝑗hasfullsupportonthehighestirrep𝜆 (𝑛) 𝑑 . Weproposethestudyofthefollowingobservablethatrespectsuch condition and only has support on the i...
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[112]
ThispreventtheuseofSimulationTechnique5 for𝑝=Ω(log𝑛)
Classical Simulation The single support observable proposed here is not a simple monomial of degree𝑝, and its decomposition into a sumofmonomialwouldrequiretoconsideralargenumberofterms. ThispreventtheuseofSimulationTechnique5 for𝑝=Ω(log𝑛). Forconstantdegree𝑝,theobservableonly...
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[113]
(18), and recalled here: E 𝑈∼𝑈(𝑚) 𝑓𝑈(𝜌,𝑂) 2 = 𝑛Õ 𝑘=0 ∥𝑃(𝑛) 𝑘 (𝜌)∥2 2∥𝑃(𝑛) 𝑘 (𝑂)∥ 2 2 𝑑(𝑛) 𝑘
Second Moment Studies: To study the second moment, we propose to verify numerically the decay of the second moment given in Eq. (18), and recalled here: E 𝑈∼𝑈(𝑚) 𝑓𝑈(𝜌,𝑂) 2 = 𝑛Õ 𝑘=0 ∥𝑃(𝑛) 𝑘 (𝜌)∥2 2∥𝑃(𝑛) 𝑘 (𝑂)∥ 2 2 𝑑(𝑛) 𝑘 . In the particular case of the single support observable...
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[114]
This difference mainly comes from the fact that Fock stateshavemainlysupportonthelastirreps(seeSectionD)whilethesingleirrepobservableonlyhassupportonthe irrep𝑝
Cause of Barren Plateaus We notice that, as for the case of number operator products in Section E, the presence of Barren Plateaus when 𝑝= Ω(log2𝑛)is due to the fact that the purity distribution of the input state{𝑃(𝑛) 𝑘 (𝜌)}𝑛 𝑘=0 is vanishing exponentially and is too small wi...
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[115]
We call𝑔𝑝({ˆ𝑛𝑖}𝑖∈𝐼), a polynomial function of number operators of degree𝑝⩽𝑛, with𝑛the number of particles
Definition and properties: In this Section, we consider a very general set of parity measurement. We call𝑔𝑝({ˆ𝑛𝑖}𝑖∈𝐼), a polynomial function of number operators of degree𝑝⩽𝑛, with𝑛the number of particles. In this Section, we focus on observables of the form: 𝑂=(−1) 𝑔𝑑({ˆ𝑛𝑖}𝑖∈𝐼...
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[116]
(A1), on an operator of the form of Eq
Irrep Decomposition: Applyingthechangeofvariable𝑆=𝑅−𝑏(|𝑏|=𝑘,𝑏⩽𝑅),the𝑘-foldapplicationoftheloweringmap𝐿,introduced in Eq. (A1), on an operator of the form of Eq. (G1) is given by: 𝐿𝑘(𝑂)= Õ 𝑆∈Φ𝑛−𝑘𝑚 𝜆𝑆|𝑆⟩⟨𝑆|,𝜆 𝑆 = Õ 𝑏∈Z𝑚 ⩾0 |𝑏|=𝑘 (−1)𝑔𝑑({𝑆𝑖+𝑏𝑖}𝑖∈𝐼) 𝑘!Î 𝑙𝑏𝑙! Ö 𝑙 (𝑆𝑙+𝑏 𝑙)! 𝑆𝑙! .(G5...
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[117]
(G3) can be efficiently simulated classi- cally
Classical Simulation Kerr gate decomposition The particular case of parity measurement observable described by Eq. (G3) can be efficiently simulated classi- cally. As highlighted in [73], the expectation value physically corresponds to that of a layer phase-shifter of the form...
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[118]
(18), and recalled here: E 𝑈∼𝑈(𝑚) 𝑓𝑈(𝜌,𝑂) 2 = 𝑛Õ 𝑘=0 ∥𝑃(𝑛) 𝑘 (𝜌)∥2 2∥𝑃(𝑛) 𝑘 (𝑂)∥ 2 2 𝑑(𝑛) 𝑘
Second Moment Studies: To study the second moment, we propose to verify numerically the decay of the second moment given in Eq. (18), and recalled here: E 𝑈∼𝑈(𝑚) 𝑓𝑈(𝜌,𝑂) 2 = 𝑛Õ 𝑘=0 ∥𝑃(𝑛) 𝑘 (𝜌)∥2 2∥𝑃(𝑛) 𝑘 (𝑂)∥ 2 2 𝑑(𝑛) 𝑘 . In order to make sure that a polynomial decay is not on...
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